Average Error: 26.5 → 8.6
Time: 3.9s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le -1.351214721549483218261281106569258966909 \cdot 10^{154} \lor \neg \left(d \le 8.000358861136643239312158095843514305232 \cdot 10^{153}\right):\\ \;\;\;\;\frac{b}{\frac{c \cdot c + d \cdot d}{c}} - \frac{a}{d + \frac{{c}^{2}}{d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\frac{{d}^{2}}{c} + c} - \frac{a}{\frac{c \cdot c + d \cdot d}{d}}\\ \end{array}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\begin{array}{l}
\mathbf{if}\;d \le -1.351214721549483218261281106569258966909 \cdot 10^{154} \lor \neg \left(d \le 8.000358861136643239312158095843514305232 \cdot 10^{153}\right):\\
\;\;\;\;\frac{b}{\frac{c \cdot c + d \cdot d}{c}} - \frac{a}{d + \frac{{c}^{2}}{d}}\\

\mathbf{else}:\\
\;\;\;\;\frac{b}{\frac{{d}^{2}}{c} + c} - \frac{a}{\frac{c \cdot c + d \cdot d}{d}}\\

\end{array}
double f(double a, double b, double c, double d) {
        double r118903 = b;
        double r118904 = c;
        double r118905 = r118903 * r118904;
        double r118906 = a;
        double r118907 = d;
        double r118908 = r118906 * r118907;
        double r118909 = r118905 - r118908;
        double r118910 = r118904 * r118904;
        double r118911 = r118907 * r118907;
        double r118912 = r118910 + r118911;
        double r118913 = r118909 / r118912;
        return r118913;
}
↓
double f(double a, double b, double c, double d) {
        double r118914 = d;
        double r118915 = -1.3512147215494832e+154;
        bool r118916 = r118914 <= r118915;
        double r118917 = 8.000358861136643e+153;
        bool r118918 = r118914 <= r118917;
        double r118919 = !r118918;
        bool r118920 = r118916 || r118919;
        double r118921 = b;
        double r118922 = c;
        double r118923 = r118922 * r118922;
        double r118924 = r118914 * r118914;
        double r118925 = r118923 + r118924;
        double r118926 = r118925 / r118922;
        double r118927 = r118921 / r118926;
        double r118928 = a;
        double r118929 = 2.0;
        double r118930 = pow(r118922, r118929);
        double r118931 = r118930 / r118914;
        double r118932 = r118914 + r118931;
        double r118933 = r118928 / r118932;
        double r118934 = r118927 - r118933;
        double r118935 = pow(r118914, r118929);
        double r118936 = r118935 / r118922;
        double r118937 = r118936 + r118922;
        double r118938 = r118921 / r118937;
        double r118939 = r118925 / r118914;
        double r118940 = r118928 / r118939;
        double r118941 = r118938 - r118940;
        double r118942 = r118920 ? r118934 : r118941;
        return r118942;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original26.5
Target0.5
Herbie8.6
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if d < -1.3512147215494832e+154 or 8.000358861136643e+153 < d

    1. Initial program 46.6

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt46.6

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
    4. Using strategy rm

    5. Applied div-sub46.6

      \[\leadsto \color{blue}{\frac{b \cdot c}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}} - \frac{a \cdot d}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]

    6. Simplified46.4

      \[\leadsto \color{blue}{\frac{b}{\frac{c \cdot c + d \cdot d}{c}}} - \frac{a \cdot d}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]

    7. Simplified45.3

      \[\leadsto \frac{b}{\frac{c \cdot c + d \cdot d}{c}} - \color{blue}{\frac{a}{\frac{c \cdot c + d \cdot d}{d}}}\]

    8. Taylor expanded around 0 16.8

      \[\leadsto \frac{b}{\frac{c \cdot c + d \cdot d}{c}} - \frac{a}{\color{blue}{d + \frac{{c}^{2}}{d}}}\]

    if -1.3512147215494832e+154 < d < 8.000358861136643e+153

    1. Initial program 18.8

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt18.8

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
    4. Using strategy rm

    5. Applied div-sub18.8

      \[\leadsto \color{blue}{\frac{b \cdot c}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}} - \frac{a \cdot d}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]

    6. Simplified17.0

      \[\leadsto \color{blue}{\frac{b}{\frac{c \cdot c + d \cdot d}{c}}} - \frac{a \cdot d}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]

    7. Simplified15.2

      \[\leadsto \frac{b}{\frac{c \cdot c + d \cdot d}{c}} - \color{blue}{\frac{a}{\frac{c \cdot c + d \cdot d}{d}}}\]

    8. Taylor expanded around 0 5.4

      \[\leadsto \frac{b}{\color{blue}{\frac{{d}^{2}}{c} + c}} - \frac{a}{\frac{c \cdot c + d \cdot d}{d}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -1.351214721549483218261281106569258966909 \cdot 10^{154} \lor \neg \left(d \le 8.000358861136643239312158095843514305232 \cdot 10^{153}\right):\\ \;\;\;\;\frac{b}{\frac{c \cdot c + d \cdot d}{c}} - \frac{a}{d + \frac{{c}^{2}}{d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\frac{{d}^{2}}{c} + c} - \frac{a}{\frac{c \cdot c + d \cdot d}{d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (- b (* a (/ d c))) (+ c (* d (/ d c)))) (/ (+ (- a) (* b (/ c d))) (+ d (* c (/ c d)))))

  (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))